4.1 Introduction

4.1.1 The Intent of Content Markup

As has been noted in the introductory section of this Recommendation, mathematics can be distinguished by its use of a (relatively)
formal language, mathematical notation. However, mathematics and its presentation should not be viewed as one and the same
thing. Mathematical sums or products exist and are meaningful to many applications completely without regard to how they are
rendered aurally or visually. The intent of the content markup in the Mathematical Markup Language is to provide an explicit
encoding of the
underlying mathematical structure of an expression, rather than any particular rendering for the expression.

There are many reasons for providing a specific encoding for content. Even a disciplined and systematic use of presentation
tags cannot properly capture this semantic information. This is because without additional information it is impossible to
decide whether a particular presentation was chosen deliberately to encode the mathematical structure or simply to achieve
a particular visual or aural effect. Furthermore, an author using the same encoding to deal with both the presentation and
mathematical structure might find a particular presentation encoding unavailable simply because convention had reserved it
for a different semantic meaning.

The difficulties stem from the fact that there are many to one mappings from presentation to semantics and vice versa. For
example the mathematical construct
"
H multiplied by
e" is often encoded using an explicit operator as in
H ×
e. In different presentational contexts, the multiplication operator might be invisible
"
He", or rendered as the spoken word
"times". Generally, many different presentations are possible depending on the context and style preferences of the author
or reader. Thus, given
"
He" out of context it may be impossible to decide if this is the name of a chemical or a mathematical product of two variables
H and
e.

Mathematical presentation also changes with culture and time: some expressions in combinatorial mathematics today have one
meaning to a Russian mathematician, and quite another to a French mathematician; see
Section 5.4.1 Notational Style Sheets for an example. Notations may lose currency, for example the use of musical sharp and flat symbols to denote maxima and minima
[Chaundy1954]. A notation in use in 1644 for the multiplication mentioned above was
He[Cajori1928].

When we encode the underlying mathematical structure explicitly, without regard to how it is presented aurally or visually,
we are able to interchange information more precisely with those systems that are able to manipulate the mathematics. In the
trivial example above, such a system could substitute values for the variables
H and
e and evaluate the result. Further interesting application areas include interactive textbooks and other teaching aids.

4.1.2 The Scope of Content Markup

The semantics of general mathematical notation is not a matter of consensus. It would be an enormous job to systematically
codify most of mathematics - a task that can never be complete. Instead, MathML makes explicit a relatively small number of
commonplace mathematical constructs, chosen carefully to be sufficient in a large number of applications. In addition, it
provides a mechanism for associating semantics with new notational constructs. In this way, mathematical concepts that are
not in the base collection of elements can still be encoded (Section 4.2.6 Syntax and Semantics).

The base set of content elements is chosen to be adequate for simple coding of most of the formulas used from kindergarten
to the end of high school in the United States, and probably beyond through the first two years of college, that is up to
A-Level or Baccalaureate level in Europe. Subject areas covered to some extent in MathML are:

arithmetic, algebra, logic and relations

calculus and vector calculus

set theory

sequences and series

elementary classical functions

statistics

linear algebra

It is not claimed, or even suggested, that the proposed set of elements is complete for these areas, but the provision for
author extensibility greatly alleviates any problem omissions from this finite list might cause.

4.1.3 Basic Concepts of Content Markup

The design of the MathML content elements are driven by the following principles:

The expression tree structure of a mathematical expression should be directly encoded by the MathML content elements.

The encoding of an expression tree should be explicit, and not dependent on the special parsing of
PCDATA or on additional processing such as operator precedence parsing.

The basic set of mathematical content constructs that are provided should have default mathematical semantics.

There should be a mechanism for associating specific mathematical semantics with the constructs.

The primary goal of the content encoding is to establish explicit connections between mathematical structures and their mathematical
meanings. The content elements correspond directly to parts of the underlying mathematical expression tree. Each structure
has an associated default semantics and there is a mechanism for associating new mathematical definitions with new constructs.

Significant advantages to the introduction of content-specific tags include:

Usage of presentation elements is less constrained. When mathematical semantics are inferred from presentation markup, processing
agents must either be quite sophisticated, or they run the risk of inferring incomplete or incorrect semantics when irregular
constructions are used to achieve a particular aural or visual effect.

It is immediately clear which kind of information is being encoded simply by the kind of elements that are used.

Combinations of semantic and presentation elements can be used to convey both the appearance and its mathematical meaning
much more effectively than simply trying to infer one from the other.

Expressions described in terms of content elements must still be rendered. For common expressions, default visual presentations
are usually clear.
"Take care of the sense and the sounds will take care of themselves" wrote Lewis Carroll
[Carroll1871]. Default presentations are included in the detailed description of each element occurring in
Section 4.4 The Content Markup Elements.

To accomplish these goals, the MathML content encoding is based on the concept of an expression tree. A content expression
tree is constructed from a collection of more primitive objects, referred to herein as
containers and
operators. MathML possesses a rich set of predefined container and operator objects, as well as constructs for combining containers
and operators in mathematically meaningful ways. The syntax and usage of these content elements and constructions is described
in the next section.

4.2 Content Element Usage Guide

Since the intent of MathML content markup is to encode mathematical
expressions in such a way that the mathematical structure of the
expression is clear, the syntax and usage of content markup must be
consistent enough to facilitate automated semantic
interpretation. There must be no doubt when, for example, an actual
sum, product or function application is intended and if specific
numbers are present, there must be enough information present to
reconstruct the correct number for purposes of computation. Of course,
it is still up to a MathML processor to decide what is to be done with such a content-based expression, and computation is
only one of many options. A renderer or a structured editor might simply use the data and its own built-in knowledge of mathematical
structure to render the object. Alternatively, it might manipulate the object to build a new mathematical object. A more computationally
oriented system might attempt to carry out the indicated operation or function evaluation.

The purpose of this section is to describe the intended, consistent usage. The requirements involve more than just satisfying
the syntactic structure specified by an XML DTD. Failure to conform to the usage as described below will result in a MathML
error, even though the expression may be syntactically valid according to the DTD.

4.2.1 Overview of Syntax and Usage

MathML content encoding is based on the concept of an expression tree. As a general rule, the terminal nodes in the tree represent
basic mathematical objects, such as numbers, variables, arithmetic operations and so on. The internal nodes in the tree generally
represent some kind of function application or other mathematical construction that builds up a compound object. Function
application provides the most important example; an internal node might represent the application of a function to several
arguments, which are themselves represented by the terminal nodes underneath the internal node.

The MathML content elements can be grouped into the following categories based on their usage:

constants and symbols

containers

operators and functions

qualifiers

relations

conditions

semantic mappings

These are the building blocks out of which MathML content expressions are constructed. Each category is discussed in a separate
section below. In the remainder of this section, we will briefly introduce some of the most common elements of each type,
and consider the general constructions for combining them in mathematically meaningful ways.

4.2.1.1 Constructing Mathematical Objects

Content expression trees are built up from basic mathematical objects. At the lowest level,
leaf nodes are encapsulated in non-empty elements that define their type. Numbers and symbols are marked by the
token elements
cn and
ci. More elaborate constructs such as sets, vectors and matrices are also marked using elements to denote their types, but rather
than containing data directly, these
container elements are constructed out of other elements. Elements are used in order to clearly identify the underlying objects. In
this way, standard XML parsing can be used and attributes can be used to specify global properties of the objects.

The containers such as
<cn>12345</cn> ,
<ci>x</ci> and
<csymbol definitionURL="mySymbol.htm" encoding="text">S</csymbol>represent mathematical numbers , identifiers and externally defined symbols. Below, we will look at
operator elements such as
plus or
sin, which provide access to the basic mathematical operations and functions applicable to those objects. Additional containers
such as
set for sets, and
matrix for matrices are provided for representing a variety of common compound objects.

For example, the number 12345 is encoded as

<cn>12345</cn>

The attributes and
PCDATA content together provide the data necessary for an application to parse the number. For example, a default base of 10 is
assumed, but to communicate that the underlying data was actually written in base 8, simply set the
base attribute to 8 as in

<cn base="8">12345</cn>

while the complex number 3 + 4i can be encoded as

<cn type="complex-cartesian">3<sep/>4</cn>

Such information makes it possible for another application to easily parse this into the correct number.

This invokes default semantics associated with the
vector element, namely an arbitrary element of a finite-dimensional vector space.

By using the
ci and
csymbol elements we have made clear that we are referring to a mathematical identifier or symbol but this does not say anything about
how it should be rendered. By default a symbol is rendered as if the
ci or
csymbol element were actually the presentation element
mi (see
Section 3.2.3 Identifier (mi)). The actual rendering of a mathematical symbol can be made as elaborate as necessary simply by using the more elaborate
presentational constructs (as described in
Chapter 3 Presentation Markup) in the body of the
ci or
csymbol element.

The default rendering of a simple
cn-tagged object is the same as for the presentation element
mn with some provision for overriding the presentation of the
PCDATA by providing explicit
mn tags. This is described in detail in
Section 4.4 The Content Markup Elements.

The issues for compound objects such as sets, vectors and matrices are all similar to those outlined above for numbers and
symbols. Each such object has global properties as a mathematical object that impact how it is to be parsed.
This may affect everything from the interpretation of operations that are applied to it to how to render
the symbols representing it. These mathematical properties are captured by setting attribute values
or by associating the properties with the object through the use of the semantics element.

4.2.1.2 Constructing General Expressions

The notion of constructing a general expression tree is essentially that of applying an operator to sub-objects. For example,
the sum
a +
b can be thought of as an application of the addition operator to two arguments
a and
b. In MathML, elements are used for operators for much the same reason that elements are used to contain objects. They are
recognized at the level of XML parsing, and their attributes can be used to record or modify the intended semantics. For example,
with the MathML
plus element, setting the
definitionURL and
encoding attributes as in

There is also another reason for using elements to denote operators. There is a crucial semantic distinction between the function
itself and the expression resulting from applying that function to zero or more arguments which must be captured. This is
addressed by making the functions self-contained objects with their own properties and providing an explicit
apply construct corresponding to function application. We will consider the
apply construct in the next section.

MathML contains many pre-defined operator elements, covering a range of mathematical subjects. However, an important class
of expressions involve unknown or user-defined functions and symbols. For these situations, MathML provides a general
csymbol element, which is discussed below.

4.2.1.3 The
apply construct

The most fundamental way of building up a mathematical expression in MathML content markup is the
apply construct. An
apply element typically applies an operator to its arguments. It corresponds to a complete mathematical expression. Roughly speaking,
this means a piece of mathematics that could be surrounded by parentheses or
"logical brackets" without changing its meaning.

For example, (x +
y) might be encoded as

<apply>
<plus/>
<ci> x </ci>
<ci> y </ci>
</apply>

The opening and closing tags of
apply specify exactly the scope of any operator or function. The most typical way of using
apply is simple and recursive. Symbolically, the content model can be described as:

<apply>
opab </apply>

where the
operands a and b are containers or other content-based elements themselves, and
op is an operator or function. Note that since
apply is a container, this allows
apply constructs to be nested to arbitrary depth.

An
apply may in principle have any number of operands:

<apply> op a b [c...] <apply>

For example, (x +
y +
z) can be encoded as

<apply>
<plus/>
<ci> x </ci>
<ci> y </ci>
<ci> z </ci>
</apply>

Mathematical expressions involving a mixture of operations result in nested occurrences of
apply. For example,
ax +
b would be encoded as

There is no need to introduce parentheses or to resort to operator precedence in order to parse the expression correctly.
The
apply tags provide the proper grouping for the re-use of the expressions within other constructs. Any expression enclosed by an
apply element is viewed as a single coherent object.

Both the function and the arguments may be simple identifiers or more complicated expressions.

In MathML 1.0 , another construction closely related to the use of the
apply element with operators and arguments was the
reln element. The
reln element was used to denote that a mathematical relation holds between its arguments, as opposed to applying an operator.
Thus, the MathML markup for the expression
x <
y was given in MathML 1.0 by:

<reln>
<lt/>
<ci> x </ci>
<ci> y </ci>
</reln>

In MathML 2.0, the
apply construct is used with all operators, including logical operators. The expression above becomes

<apply>
<lt/>
<ci> x </ci>
<ci> y </ci>
</apply>

in MathML 2.0. The use of
reln with relational operators is supported
for reasons of backwards compatibility, but deprecated. Authors creating new content are
encouraged to use
apply in all cases.

4.2.1.4 Explicitly defined functions and operators

The most common operations and functions such as
plus and
sin have been predefined explicitly as empty elements (see
Section 4.4 The Content Markup Elements).
The definitionURL attribute can be used by the author to record
that a different sort of algebraic operation is intended. This allows essentially the same notation to be re-used for a
discussion taking place in a different algebraic domain.

Due to the nature of mathematics the notation must be extensible. The key to extensibility is the ability of the user to define
new functions and other symbols to expand the terrain of mathematical discourse.

It is always possible to create arbitrary expressions, and then to use them as symbols in the language. Their properties can
then be inferred directly from that usage as was done in the previous section. However, such an approach would preclude being
able to encode the fact that the construct was a known symbol, or to record its mathematical properties except by actually
using it. The
csymbol element is used as a container to construct a new symbol in much the same way that
ci is used to construct an identifier. (Note that
"symbol" is used here in the abstract sense and has no connection with any presentation of the construct on screen or paper).
The difference in usage is that
csymbol should refer to some mathematically defined concept with an external definition referenced via the
definitionURL attribute, whereas
ci is used for identifiers that are essentially
"local" to the MathML expression
. The target of the
definitionURL attribute on the
csymbol element may encode the definition in any format; the particular encoding in use is given by the encoding attribute.
In contrast, the definitionURL attribute on a
ci element, might be used to associate an identifier with another
sub-expression by referring to its id attribute.
This approach can be used, for example to indicate clearly that a particular
ci element is an instance of a ci element that has been declared to have some properties using the declare construct (see Section 4.4.2.8 Declare (declare))
or that it is an instance of a specific bound variable as declared
by a use of the bvar (see Section 4.4.5.6 Bound variable (bvar)) element.

To use
csymbol to describe a completely new function, we write for example

The
definitionURL attribute specifies a URI that provides a written definition for the
Christoffel symbol. Suggested default definitions for the content elements of MathML appear in
Appendix C Content Element Definitions in a format based on OpenMath, although there is no requirement that a particular format be used. The role of the
definitionURL attribute is very similar to the role of definitions included at the beginning of many mathematical papers, and which often
just refer to a definition used by a particular book.

MathML 1.0 supported the use of the
fn to encode the fact that a construct is explicitly being used as a function or operator. To record the fact that
F+
G is being used semantically as if it were a function, it was encoded as:

<fn>
<apply>
<plus/>
<ci>F</ci>
<ci>G</ci>
</apply>
</fn>

This usage, although allowed in MathML 2.0 for reasons of backwards compatibility,
is now deprecated.
The fact that a construct is being used as an operator is clear from the position of the construct as the
first child of the
apply. If it is required to add additional information to the construct, it should be wrapped in a
semantics element, for example:

MathML 1.0 supported the use of
definitionURL with
fn to refer to external definitions for user-defined
functions. This usage, although allowed for reasons of backwards
compatibility, is deprecated in
MathML 2.0 in favor of using
csymbol to define the function, and then
apply to link the function to its arguments. For example:

4.2.1.5 The inverse construct

Given functions, it is natural to have functional inverses. This is handled by the
inverse element.

Functional inverses can be problematic from a mathematical point of view in that they implicitly involve the definition of
an inverse for an arbitrary function
F. Even at the K-through-12 level the concept of an inverse
F-1 of many common functions
F is not used in a uniform way. For example, the definitions used for the inverse trigonometric functions may differ slightly
depending on the choice of domain and/or branch cuts.

MathML adopts the view: if
F is a function from a domain
D to
D', then the inverse
G of
F is a function over
D' such that
G(F(x)) =
x for
x in
D. This definition does not assert that such an inverse exists for all or indeed any
x in
D, or that it is single-valued anywhere. Also, depending on the functions involved, additional properties such as
F(G(y)) =
y for
y in
D' may hold.

The
inverse element is applied to a function whenever an inverse is required. For example, application of the inverse sine function to
x, i.e. sin-1 (x), is encoded as:

<apply>
<apply> <inverse/> <sin/> </apply>
<ci> x </ci>
</apply>

While
arcsin is one of the predefined MathML functions, an explicit reference to sin-1(x) might occur in a document discussing possible definitions of
arcsin.

4.2.1.6 The declare construct

Consider a document discussing the vectors
A = (a, b,
c) and
B = (d, e,
f), and later including the expression
V =
A +
B. It is important to be able to communicate the fact that wherever
A and
B are used they represent a particular vector. The properties of that vector may determine aspects of operators such as
plus.

The simple fact that
A is a vector can be communicated by using the markup

<ci type="vector">A</ci>

but this still does not communicate, for example, which vector is involved
or its dimensions.

The declare construct is used to associate
specific properties or meanings with an object. The actual declaration
itself is not rendered visually (or in any other form). However, it
indirectly impacts the semantics of all affected uses of the declared
object.

Declarations must occur at the beginning of a math
element. The scope of a declaration is the entire
math element in which the declaration is made.
The scope attribute of a declare
may be included but has no effect since the two possible values of
"local" or "global"
now have the same meaning. The "global" attribute value
is still allowed for backwards compatibility with MathML 1.0.,
but is deprecated in MathML 2.0.

The uses of the declare element range from
resetting default attribute values to associating an expression with a
particular instance of a more elaborate structure. Subsequent uses of the
original expression (within the scope of the declare) play the same semantic role as would the
paired object.

remains unchanged but the expression can be interpreted properly as vector addition.

There is no requirement to declare an expression to stand for a specific object. For example, the declaration

<declare type="vector">
<ci> A </ci>
</declare>

specifies that
A is a vector without indicating the number of components or the values of specific components.
Any attribute which is valid for the target element can be assigned in this way,
with the possible values being the same as would ordinarily be assigned to such an object.

4.2.1.7 The lambda construct

The lambda calculus allows a user to construct a function from a variable and an expression. For example, the lambda construct
underlies the common mathematical idiom illustrated here:

Let
f be the function taking
x to
x2 + 2

There are various notations for this concept in mathematical literature, such as
(x,
F(x)) = F or
(x,
[F]) =F, where x is a free variable in F.

This concept is implemented in MathML with the lambda element. A lambda construct with n
(possibly 0) internal variables is encoded by a lambda
element, where the first n children are bvar elements
containing the identifiers of the internal variables. This is followed by an
optional domainofapplication qualifier and an expression defining the
function. The defining expression is typically an apply, but can also be
any expression.

4.2.1.8 The use of qualifier elements

The last example of the preceding section illustrates the use of
qualifier elements
lowlimit,
uplimit, and
bvar in conjunction with the
int element. A number of common mathematical constructions involve additional data that is either
implicit in conventional notation, such as a bound variable, or thought of as part of the operator
rather than an argument, as is the case with the limits of a definite integral.

Content markup uses qualifier elements in conjunction with a number of operators, including integrals,
sums, series, and certain differential operators.
They may also be used by user defined functions such
as those added by making use of the csymbol element, or by use of lambda expressions.
Qualifier elements appear in the same
apply element with one of these operators. In general, they must appear in a
certain order, and their precise meaning depends on the operators being used. For details
about the use of qualifiers with the predefined operators see
Section 4.2.3.2 Operators taking Qualifiers. The role of qualifiers for
user defined functions is determined solely by the definition of each function.

A typical use of a qualifier is to identify a bound variable through use of
the bvar element, or to
restrict the values of the bound variable to a particular domain of application or in
some other way. For example, a domain of application can be given explicitly using
the domainofapplication element or by restricting the values of the bvar element
to an interval or by conditions. A
condition element can be used to place restrictions directly on the on a bound variable.
This allows MathML to define sets by rule, rather than enumeration.
The following markup, for instance, encodes the set {x |
x < 1}:

Another typical use is the "lifting" of
n-ary operators to "big operators", for instance
the n-ary union operator to the union operator over sets, as the
union of the U-complements over a family F of sets in
this construction

This general construction gives natural lifted version of many
n-ary operators, such as plus, times,
max, min, gcd, lcm, mean,
sdev, variance, median, mode,
and, or, xor, union,
intersect, cartesianproduct. The meaning of expressions of
the first form is that the operator is applied to the values of the
expression in the last child (where the bound variables vary as specified
in the qualifiers). The meaning of constructions of the second form is that
the function in the last child is applied to all the values of the set in
the domainofapplication qualifier.

4.2.1.9 Rendering of Content elements

While the primary role of the MathML content element set is to directly encode the mathematical structure of expressions independent
of the notation used to present the objects, rendering issues cannot be ignored. Each content element has a default rendering,
given in
Section 4.4 The Content Markup Elements, and several mechanisms (including
Section 4.3.3.2 General Attributes) are provided for associating a particular rendering with an object.

4.2.2 Containers

Containers provide a means for the construction of mathematical objects of a given type.

The
cn element is the MathML token element used to represent numbers. The supported types of numbers include:
"real",
"integer",
"rational",
"complex-cartesian", and
"complex-polar", with
"real" being the default type. An attribute
base (with default value
"10") is used to help specify how the content is to be parsed. The content itself is essentially
PCDATA, separated by
<sep/> when two parts are needed in order to fully describe a number. For example, the real number 3 is constructed by
<cn type="real"> 3 </cn>, while the rational number 3/4 is constructed as
<cn type="rational"> 3<sep/>4 </cn>. The detailed structure and specifications are provided in
Section 4.4.1.1 Number (cn).

ci

The
ci element, or
"content identifier" is used to construct a variable, or an identifier. A
type attribute indicates the type of object the symbol represents. Typically,
ci represents a real scalar, but no default is specified. The content is either
PCDATA or a general presentation construct (see
Section 3.1.6 Summary of Presentation Elements). For example,

<ci>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</ci>

encodes an atomic symbol that displays visually as
c1
which, for purposes of content, is treated as a single symbol representing a real number.
The definitionURL attribute can be used to
identify special properties or to refer to
a defining instance of (for example) a bound variable.
The detailed structure and specifications are provided in
Section 4.4.1.2 Identifier (ci).

csymbol

The
csymbol element, or
"content symbol" is used to construct a symbol whose semantics are not part of the core content elements provided by MathML,
but defined
outside of the MathML specification.
csymbol does not make any attempt to describe how to map the arguments occurring in any application of the function into a new MathML
expression. Instead, it depends on its
definitionURL attribute to point to a particular meaning, and the
encoding attribute to give the syntax of this definition. The content of a
csymbol is either
PCDATA or a general presentation construct (see
Section 3.1.6 Summary of Presentation Elements). For example,

encodes an atomic symbol that displays visually as
C2 and that, for purposes of content, is treated as a single symbol representing the space of twice-differentiable continuous
functions. The detailed structure and specifications are provided in
Section 4.4.1.3 Externally defined symbol (csymbol).

4.2.2.2 Constructors

MathML provides a number of elements for combining elements into familiar compound objects. The compound objects include things
like lists and sets. Each constructor produces a new type of object.

interval

The
interval element is described in detail in
Section 4.4.2.4 Interval (interval). It denotes an interval on the real line with the values represented by its children as end points. The
closure attribute is used to qualify the type of interval being represented. For example,

The
set and
list elements are described in detail in
Section 4.4.6.1 Set (set) and
Section 4.4.6.2 List (list). Typically, the child elements of a possibly empty
list element are the actual components of an ordered
list. For example, an ordered list of the three symbols
a,
b, and
c is encoded as

<list> <ci> a </ci> <ci> b </ci> <ci> c </ci> </list>

Sets and lists can also be constructed by evaluating a function over a domain of
application, each evaluation corresponding to a term of the set or list. In the most
general form a domain is explicitly specified by
a domainofapplication element together with optional bvar elements.
Qualifications involving a domainofapplication element can be abbreviated
in several ways as described in Section 4.2.3.2 Operators taking Qualifiers. For example, a bvar and a
condition element can be used to define lists
where membership depends on satisfying certain conditions.

An
order attribute can be used to specify what ordering is to be used. When the nature of the child elements permits, the ordering
defaults to a numeric or lexicographic ordering.

Sets are structured much the same as lists except that there is no implied ordering and the
type of set may be
"normal" or
"multiset" with
"multiset" indicating that repetitions are allowed.

For both sets and lists, the child elements must be valid MathML content elements. The type of the child elements is not restricted.
For example, one might construct a list of equations, or of inequalities.

matrix and matrixrow

The
matrix element is used to represent mathematical matrices. It is described in detail in
Section 4.4.10.2 Matrix (matrix). It has zero or more child elements, all of which are
matrixrow elements. These in turn expect zero or more child elements that evaluate to algebraic expressions or numbers. These sub-elements
are often real numbers, or symbols as in

The
matrixrow elements must always be contained inside of a matrix, and all rows in a given matrix must have the same number of elements.

Note that the behavior of the
matrix and
matrixrow elements is substantially different from the
mtable and
mtr presentation elements.

A matrix can also be constructed by evaluating a bivariate function over a specific domain of
application, each evaluation corresponding to an entry in the matrix. In its most
general form a domain of application is explicitly specified by
a domainofapplication element and a function which when evaluated at points of the domain
produces entries in the matrix. Optionally the domainofapplication
can be augmented by bvar elements and an
algebraic expression expressed in terms of them.
Qualifications defined by a domainofapplication element can be abbreviated
in several ways as described in Section 4.2.3 Functions, Operators and Qualifiers.

vector

The
vector element is described in detail in
Section 4.4.10.1 Vector (vector). It constructs vectors from an
n-dimensional vector space so that its
n child elements typically represent real or complex valued scalars as in the three-element vector

A vector can also be constructed by evaluating a function over a specific domain of
application, each evaluation corresponding to an entry in the vector. In its most
general form a domain is explicitly specified by
a domainofapplication element and a function. Optionally the domainofapplication
can be augmented by a bvar element and an
algebraic expression expressed in terms of it.
Qualifications defined by a domainofapplication element can be abbreviated
in several ways as described in Section 4.2.3 Functions, Operators and Qualifiers.

apply

The
apply element is described in detail in
Section 4.4.2.1 Apply (apply). Its purpose is to apply a function or operator to its arguments to produce an expression representing an element of the
codomain of the function. It is involved in everything from forming sums such as
a +
b as in

<apply>
<plus/>
<ci> a </ci>
<ci> b </ci>
</apply>

through to using the sine function to construct sin(a) as in

<apply>
<sin/>
<ci> a </ci>
</apply>

or constructing integrals. Its usage in any particular setting is determined largely by the properties of the function (the
first child element) and as such its detailed usage is covered together with the functions and operators in
Section 4.2.3 Functions, Operators and Qualifiers.

MathML 2.0 takes the view that this should be regarded as the application of a Boolean function, and as such could be constructed
using
apply. The use of
reln with logical operators is supported
for reasons of backwards compatibility, but deprecated in favor of
apply.

fn

The
fn element was used in MathML 1.0 to make
explicit the fact that an expression is being used as a function or
operator. This is allowed in MathML 2.0 for backwards compatibility,
but is deprecated, as the use of
an expression as a function or operator is clear from its position as
the first child of an
apply.
fn is discussed in detail in
Section 4.4.2.3 Function (fn).

lambda

The
lambda element is used to construct a user-defined function from an expression and zero or more free variables. The lambda construct
with
n internal variables takes
n+1 children. The first (second, up to
n) is a
bvar containing the identifiers of the internal variables. The last is an expression defining the function. This is typically
an
apply, but can also be any container element. The following constructs
(x, sin
x)

4.2.2.3 Special Constructs

The
declare construct is described in detail in
Section 4.4.2.8 Declare (declare). It is special in that its entire purpose is to modify
the semantics of other objects. It is not rendered visually or aurally.

The need for declarations arises any time a symbol (including more general presentations)
is being used to represent an instance of an object of a particular type. For example,
you may wish to declare that the symbolic identifier V represents a vector.
The single argument form can be used to set properties of objects by setting
the default values of implied attribute values to specific values.

The declaration

<declare type="vector"><ci>V</ci></declare>

resets the default type attribute of
<ci>V</ci> to
"vector" for all affected occurrences of
<ci>V</ci>. This avoids having to write
<ci type="vector">V</ci> every time you use the symbol.

More generally,
declare can be used to associate expressions with specific content. For example, the declaration

associates the symbol
F with a new function defined by the
lambda construct. Within the scope where the declaration is in effect, the expression

<apply>
<ci>F</ci>
<ci> U </ci>
</apply>

stands for the integral of
U from 0 to
a.

The
declare element can also be used to change the definition of a function or operator. For example, if the URL
http://.../MathML:noncommutplus described a non-commutative plus operation encoded in Maple syntax, then the declaration

From the point of view of usage, MathML regards functions (for example
sin and
cos) and operators (for example
plus and
times) in the same way. MathML predefined functions and operators are all canonically empty elements.

Note that the
csymbol element can be used to construct a user-defined symbol that can be used as a function or operator.

4.2.3.1 Predefined functions and operators

MathML functions can be used in two ways. They can be used as the operator within an
apply element, in which case they refer to a function evaluated at a specific value. For example,

<apply>
<sin/>
<cn>5</cn>
</apply>

denotes a real number, namely sin(5).

MathML functions can also be used as arguments to other operators, for example

<apply>
<plus/><sin/><cos/>
</apply>

denotes a function, namely the result of adding the sine and cosine functions in some function space. (The default semantic
definition of
plus is such that it infers what kind of operation is intended from the type of its arguments.)

The number of child elements in the
apply is defined by the element in the first (i.e. operator) position.

Unary operators are followed by exactly one other child element within the
apply.

Binary operators are followed by exactly two child elements.

N-ary operators are followed by any number of child elements.
Alternatively, their operands may be generated by allowing a function or expression to vary over a domain
of application.

Some operators have multiple classifications depending on how they are used. For example the
minus operator can be both unary and binary.

Operators taking qualifiers are canonically empty functions that differ from ordinary empty functions only in that they support
the use of special
qualifier elements to specify their meaning more fully. They are used in exactly the same way as ordinary operators, except that when
they are used as operators, certain qualifier elements are also permitted to be in the enclosing
apply. Qualifiers always follow the operator and precede any arguments that are present. If more than one qualifier is present,
they appear in the order
bvar,
lowlimit,
uplimit,
interval,
condition,
domainofapplication,
degree,
momentabout,
logbase. A typical example is:

The (lowlimit,uplimit) pair, the interval and the condition are all shorthand notations
specifying a particular domain of application and should not be used if domainofapplication is used.
These shorthand notations are provided as they correspond to common usage cases and map more easily to familiar presentations.
For example, the lowlimit, uplimit pair can be used where explicit upper and
lower limits and a bound variable are all known, while an interval can be used in the same situation
but without an explicit bound variable as in:

The condition qualifier corresponds to situations where the domain of application is a set described by
simple conditions placed directly on the bound variable(s). (Such conditions are often displayed in place of a lower bound.)
An example of the use of condition is:

The most general qualifier is the domainofapplication.
It is used to provide the name of or a description of
the set over which the operation is to take place and should be used explicitly whenever there is
danger of confusing the role one of the short forms such as in an expression with
multiple interval elements. It can be used to write an expression for the integral a function over a named set
as in

Note that the order of bound variables of the integral must correspond to that used by
the set constructor in the domainofapplication.

By using the deprecated fn element, it was possible to associate a qualifier schema with a function
before it was applied to an argument. For example, a function acting on integrable functions on the interval [0,1]
could have been written:

This second form has the advantage of making the intended meaning explicit.

In addition to the defined usage in MathML, qualifier schemata may be used with any user-defined symbol
(e.g. using csymbol) or construct. The meaning of such a usage is not defined by MathML;
it would normally be user-defined using the definitionURL attribute.

The meaning and usage of qualifier schemata varies from function to function. The following list summarizes the usage of qualifier
schemata with the MathML functions taking qualifiers.

int

The
int function accepts the
lowlimit,
uplimit,
bvar,
interval,
condition and
domainofapplication schemata. If both
lowlimit and
uplimit schemata are present, they denote the limits of a definite integral. The domain of integration may alternatively be specified
using
interval,
condition or
domainofapplication. The
bvar schema signifies the variable of integration.

diff

The
diff function accepts the
bvar schema. The
bvar schema specifies with respect to which variable the derivative is being taken. The
bvar may itself contain a
degree schema that is used to specify the order of the derivative, i.e. a first derivative, a second derivative, etc. For example,
the second derivative of
f with respect to
x is:

The
partialdiff operator accepts zero or more
bvar schemata, and an optional degree qualifier schema. The
bvar schema specify, in order, the variables with respect to which the derivative is being taken. Each
bvar element may contain a
degree schema which is used to specify the order of the derivative being taken with respect to that
variable. The optional degree schema qualifier associated with the
partialdiff element itself (that is, appearing as a child of the enclosing
apply element rather than of one of the bvar qualifiers) is used to represent
the total degree of the differentiation. Each
degree schema used with partialdiff is expected
to contain a single child schema. For example,

The
sum and
product functions accept the
bvar,
lowlimit,
uplimit,
interval,
condition and
domainofapplication schemata. If both
lowlimit and
uplimit schemata are present, they denote the limits of the sum or product. The limits may alternatively be specified using the
interval,
condition or
domainofapplication schema. The
bvar schema signifies the internal variable in the sum or product. A typical example might be:

When used with
sum or
product, each qualifier schema is expected to contain a single child schema; otherwise an error is generated.

limit

The
limit function accepts zero or more
bvar schemata, and optional
condition and
lowlimit schemata. A
condition may be used to place constraints on the
bvar. The
bvar schema denotes the variable with respect to which the limit is being taken. The
lowlimit schema denotes the limit point. When used with
limit, the
bvar and
lowlimit schemata are expected to contain a single child schema; otherwise an error is generated.

log

The
log function accepts only the
logbase schema. If present, the
logbase schema denotes the base with respect to which the logarithm is being taken. Otherwise, the log is assumed to be base 10.
When used with
log, the
logbase schema is expected to contain a single child schema; otherwise an error is generated.

moment

The
moment function accepts the
degree and momentabout schema. If present, the
degree schema denotes the order of the moment. Otherwise, the moment is assumed to be the first order moment. When used with
moment, the
degree schema is expected to contain a single child schema; otherwise an error is generated. If present, the
momentabout schema denotes the point about which the moment is taken. Otherwise, the moment is assumed to be the moment about zero.

min, max

The
min and
max functions accept a
bvar schema in cases where the maximum or minimum is being taken over a set of values specified by a
condition schema together with an expression to be evaluated on that set.
In MathML1.0, the
bvar element was optional when using a
condition; if a
condition element containing a single variable was given by itself following a
min or
max operator, the variable was implicitly
assumed to be bound, and the expression to be maximized or minimized
(if absent) was assumed to be the single bound variable. This usage
is deprecated in MathML 2.0 in
favor of explicitly stating the bound variable(s) and the expression
to be maximized or minimized in all cases.

The universal and existential quantifier operators
forall and
exists are used in conjunction with one or more
bvar schemata to represent simple logical assertions. There are two ways of using the logical quantifier operators. The first
usage is for representing a simple, quantified assertion. For example, the statement
"there exists
x < 9" would be represented as:

The MathML content tags include a number of canonically empty elements which denote arithmetic and logical relations. Relations
are characterized by the fact that, if an external application were to evaluate them (MathML does not specify how to evaluate
expressions), they would typically return a truth value. By contrast, operators generally return a value of the same type
as the operands. For example, the result of evaluating
a <
b is either true or false (by contrast, 1 + 2 is again a number).

Relations are bracketed with their arguments using the
apply element in the same way as other functions. In MathML 1.0, relational operators were bracketed using
reln. This usage, although still supported,
is now deprecated in favor of
apply. The element for the relational operator is the first child element of the
apply. Thus, the example from the preceding paragraph is properly marked up as:

<apply>
<lt/>
<ci>a</ci>
<ci>b</ci>
</apply>

The number of child elements in the
apply is defined by the element in the first (i.e. relation) position.

Unary relations are followed by exactly one other child element within the
apply.

Binary relations are followed by exactly two child elements.

N-ary relations are followed by zero or more child elements.

Some elements have more than one such classification. For example,
the minus element is both unary and binary.

4.2.5 Conditions

condition

condition

The condition element is used to assert that a Boolean valued expression should be true.
When used in an an apply element to place a condition on a bound variable, it forms a
shorthand notation for specifying a domain of application (see Section 4.4.2.15 Domain of Application (domainofapplication))
since it restricts the permissible values for that bound variable.
In the context of quantifier operators, this corresponds to the "such that" construct used
in mathematical expressions. As a shorthand for domainofapplication it is used
in conjunction with operators like int and sum, or to specify argument lists
for operators like min and max.

A condition element contains a single child that is either an
apply, a
reln element (deprecated), or a set (deprecated)
indicating membership in that set.
Compound conditions are indicated by applying relations such as and inside the child of the condition.

4.2.6 Syntax and Semantics

mappings

semantics,
annotation,
annotation-xml

The use of content markup rather than presentation markup for mathematics is sometimes referred to as
semantic tagging[Buswell1996]. The parse-tree of a valid element structure using MathML content elements corresponds directly to the expression tree of
the underlying mathematical expression. We therefore regard the content tagging itself as encoding the
syntax of the mathematical expression. This is, in general, sufficient to obtain some rendering and even some symbolic manipulation
(e.g. polynomial factorization).

However, even in such apparently simple expressions as
X +
Y, some additional information may be required for applications such as computer algebra. Are
X and
Y integers, or functions, etc.?
"Plus" represents addition over which field? This additional information is referred to as
semantic mapping. In MathML, this mapping is provided by the
semantics,
annotation and
annotation-xml elements.

The
semantics element is the container element for the MathML expression together with its semantic mappings.
semantics expects a variable number of child elements. The first is the element (which may itself be a complex element structure) for
which this additional semantic information is being defined. The second and subsequent children, if any, are instances of
the elements
annotation and/or
annotation-xml.

The
semantics element also accepts the
definitionURL and
encoding attributes for use by external processing applications. One use might be a URI for a semantic content dictionary, for example.
Since the semantic mapping information might in some cases be provided entirely by the
definitionURL attribute, the
annotation or
annotation-xml elements are optional.

The
annotation element is a container for arbitrary data. This data may be in the form of text, computer algebra encodings, C programs,
or whatever a processing application expects.
annotation has an attribute
"encoding" defining the form in use. Note that the content model of
annotation is
PCDATA, so care must be taken that the particular encoding does not conflict with XML parsing rules.

The
annotation-xml element is a container for semantic information in well-formed XML. For example, an XML form of the OpenMath semantics could
be given. Another possible use here is to embed, for example, the presentation tag form of a construct given in content tag
form in the first child element of
semantics (or vice versa).
annotation-xml has an attribute
"encoding" defining the form in use.

where
OMA is the element defining the additional semantic information.

Of course, providing an explicit semantic mapping at all is optional, and in general would only be provided where there is
some requirement to process or manipulate the underlying mathematics.

4.2.7 Semantic Mappings

Although semantic mappings can easily be provided by various proprietary, or highly specialized encodings, there are no widely
available, non-proprietary standard schemes for semantic mapping. In part to address this need, the goal of the OpenMath effort
is to provide a platform-independent, vendor-neutral standard for the exchange of mathematical objects between applications.
Such mathematical objects include semantic mapping information. The OpenMath group has defined an XML syntax for the encoding
of this information
[OpenMath2000]. This element set could provide the basis of one
annotation-xml element set.

An attractive side of this mechanism is that the OpenMath syntax is specified in XML, so that a MathML expression together
with its semantic annotations can be validated using XML parsers.

4.2.8 Constants and Symbols

MathML provides a collection of predefined constants and symbols which represent frequently-encountered concepts in K-12 mathematics.
These include symbols for well-known sets, such as
integers and
rationals, and also some widely known constant symbols such as
false,
true,
exponentiale.

4.2.9 MathML element types

MathML functions, operators and relations can all be thought of as mathematical functions if viewed in a sufficiently abstract
way. For example, the standard addition operator can be regarded as a function mapping pairs of real numbers to real numbers.
Similarly, a relation can be thought of as a function from some space of ordered pairs into the set of values {true, false}.
To be mathematically meaningful, the domain and codomain of a function must be precisely specified. In practical terms, this
means that functions only make sense when applied to certain kinds of operands. For example, thinking of the standard addition
operator, it makes no sense to speak of
"adding" a set to a function. Since MathML content markup seeks to encode mathematical expressions in a way that can be unambiguously
evaluated, it is no surprise that the types of operands is an issue.

MathML specifies the types of arguments in two ways. The first way is by providing precise instructions for processing applications
about the kinds of arguments expected by the MathML content elements denoting functions, operators and relations. These operand
types are defined in a dictionary of default semantic bindings for content elements, which is given in
Appendix C Content Element Definitions. For example, the MathML content dictionary specifies that for real scalar arguments the plus operator is the standard commutative
addition operator over a field. The elements
cn has a
type attribute with a default value of
"real". Thus some processors will be able to use this information to verify the validity of the indicated operations.

Although MathML specifies the types of arguments for functions, operators and relations, and provides a mechanism for typing
arguments, a MathML processor is not required to do any type checking. In other words, a MathML processor will not generate
errors if argument types are incorrect. If the processor is a computer algebra system, it may be unable to evaluate an expression,
but no MathML error is generated.

4.3 Content Element Attributes

4.3.1 Content Element Attribute Values

Content element attributes are all of the type
CDATA, that is, any character string will be accepted as valid. In addition, each attribute has a list of predefined values, which
a content processor is expected to recognize and process. The reason that the attribute values are not formally restricted
to the list of predefined values is to allow for extension. A processor encountering a value (not in the predefined list)
which it does not recognize may validly process it as the default value for that attribute.

4.3.2 Attributes Modifying Content Markup Semantics

Each attribute is followed by the elements to which it can be applied.

4.3.2.2
closure

4.3.2.3
definitionURL

csymbol, declare, semantics, any operator element

points to an external definition of the semantics of the symbol or construct being declared. The value is a URL or URI that
should point to some kind of definition. This definition overrides the MathML default semantics.

At present, MathML does not specify the format in which external semantic definitions should be given. In particular,
there is no requirement that the target of the URI be loadable and parseable.
An external definition could, for example, define the semantics in human-readable form.

Ideally, in most situations the definition pointed to by the
definitionURL attribute would be some standard, machine-readable format. However, there are reasons why MathML does not require such a
format.

No such format currently exists. There are several projects underway
to develop and implement standard semantic encoding formats,
most notably the OpenMath effort.
By nature, the development of a comprehensive system of semantic encoding
is a very large enterprise, and while much work has been done, much
additional work remains. Even though the
definitionURL is designed and intended for use
with a formal semantic encoding language such as OpenMath, it is premature
to require any one particular format.

There will always be situations where some non-standard format is preferable.
This is particularly true in situations where authors are describing new
ideas.
It is anticipated that in the near term, there will be a variety
of renderer-dependent implementations of the
definitionURL attribute.

A translation tool might simply prompt the user with the specified definition
in situations where the proper semantics have been overridden, and in this
case, human-readable definitions will be most useful.

Other software may utilize OpenMath encodings.

Still other software may use proprietary encodings, or look for definitions
in any of several formats.

As a consequence, authors need to be aware that there is no guarantee a generic renderer will be able to take advantage of
information pointed to by the
definitionURL attribute. Of course, when widely-accepted standardized semantic encodings are available, the definitions pointed to can
be replaced without modifying the original document. However, this is likely to be labor intensive.

There is no default value for the
definitionURL attribute, i.e. the semantics are defined
within the MathML fragment, and/or by the MathML default semantics.

4.3.2.4
encoding

annotation, annotation-xml, csymbol, semantics, all operator elements

indicates the encoding of the annotation, or in the case of
csymbol ,
semantics and operator elements, the syntax of the target referred to by
definitionURL. Predefined values are
"MathML-Presentation",
"MathML-Content". Other typical values:
"TeX",
"OpenMath". Note that this is unrelated to the text
encoding of the document as specified for example in the encoding
pseudo-attribute of an XML declaration.

The default value is "", i.e. unspecified.

4.3.2.5
nargs

declare

indicates number of arguments for function declarations. Pre-defined values:
"nary", or any numeric string.

4.3.2.8
scope

In MathML 2.0, a declare has been restricted to occur only at the beginning of a
math element. Thus, there is no difference between
the two possible scope values and the scope attribute may be
safely ignored.
The "global" attribute value has been
deprecated for this role
as "local" better represents the concept.
Ideally, one would like to make document-wide declarations by setting the value of the
scope attribute to be
"global-document". However, the proper mechanism for document-wide declarations very much depends on details of the way in
which XML will be embedded in HTML, future XML style sheet mechanisms, and the underlying Document Object Model.

Since these supporting technologies are still in flux at present, the MathML specification does not include
"global-document" as a pre-defined value of the
scope attribute. It is anticipated, however, that this issue will be revisited in future revisions of MathML as supporting technologies
stabilize. In the near term, MathML implementors that wish to simulate the effect of a document-wide declaration are encouraged
to pre-process documents in order to distribute document-wide declarations to each individual
math element in the document.

4.3.2.9
type

Note: Each data type implies that the data adheres to certain formatting conventions, detailed below. If the data fails to
conform to the expected format, an error is generated. Details of the individual formats are:

real

A real number is presented in decimal notation. Decimal notation consists of an optional sign
("+" or
"-") followed by a string of digits possibly separated into an integer and a fractional part by a
"decimal point". Some examples are 0.3, 1, and -31.56. If a different
base is specified, then the digits are interpreted as being digits computed to that base.

e-notation

A real number may also be presented in scientific notation. Such numbers have two parts (a mantissa and an exponent) separated
by
sep. The first part is a real number, while the second part is an integer exponent indicating a power of the base.
For example, 12.3<sep/>5 represents
12.3 times 105.
The default presentation of this example is 12.3e5.

integer

An integer is represented by an optional sign followed by a string of 1 or more
"digits". What a
"digit" is depends on the
base attribute. If
base is present, it specifies the base for the digit encoding, and it specifies it base 10. Thus
base='16' specifies a hex encoding. When
base > 10, letters are added in alphabetical order as digits. The legitimate values for
base are therefore between 2 and 36.

rational

A rational number is two integers separated by
<sep/>. If
base is present, it specifies the base used for the digit encoding of both integers.

complex-cartesian

A complex number is of the form two real point numbers separated by
<sep/>.

complex-polar

A complex number is specified in the form of a magnitude and an angle (in radians). The raw data is in the form of two real
numbers separated by
<sep/>.

constant

The
"constant" type is used to denote named constants. Several important constants such as
pi have been included explicitly in MathML 2.0 as empty elements.
This use of the cn is discouraged in favor of the defined constants, or the use of
csymbol with an appropriate value for the definitionURL.
For example, instead of using the pi element, an instance of
<cn type="constant">&pi;</cn> could be used. This
should be interpreted as having the semantics of the mathematical constant Pi. The data for a constant
cn tag may be one of the following common constants:

Symbol

Value

&pi;

The usual
&pi; of trigonometry: approximately 3.141592653...

&ExponentialE; (or
&ee;)

The base for natural logarithms: approximately 2.718281828...

&ImaginaryI; (or
&ii;)

Square root of -1

&gamma;

Euler's constant: approximately 0.5772156649...

&infin; (or
&infty;)

Infinity. Proper interpretation varies with context

&true;

the logical constant
true

&false;

the logical constant
false

&NotANumber; (or
&NaN;)

represents the result of an ill-defined floating point division

ci

indicates type of the identifier. Predefined values:
"integer",
"rational",
"real",
"complex",
"complex-polar",
"complex-cartesian",
"constant",
"function" or the name of any content element. The meanings of the attribute values shared with
cn are the same as those listed for the cn element.
The attribute value "complex" is intended for use when an identifier
represents a complex number but the particular representation (such as polar or cartesian) is
either not known or is irrelevant.

The default value is "", i.e. unspecified.

declare

indicates a type value that is to be attached to the first child of the declare.
The first child of the declare must accept a type attribute and the attribute value provided must be appropriate for that element. For example, if the first child is a ci
element then the attribute value must be valid for a ci element.

is used to capture the notion of one quantity approaching another. It occurs as a
container so that it can more easily be used in the construction of a limit expression.
Predefined values:
"above",
"below",
"two-sided".

The default value is
"two-sided".

4.3.3 Attributes Modifying Content Markup Rendering

4.3.3.1
type

The
type attribute, in addition to conveying semantic information, can be interpreted to provide rendering information. For example
in

<ci type="vector">V</ci>

a renderer could display a bold
V for the vector.

4.3.3.2 General Attributes

All content elements support the following general attributes that can be used to modify the rendering of the markup.

Content or semantic tagging goes along with the (frequently implicit) premise
that, if you know the semantics, you can always work out a presentation form. When
an author's main goal is to mark up re-usable, mathematical expressions that
can be evaluated, the exact rendering of the expression is probably not critical, provided that it is easily understandable.
However, when an author's goal is more along the lines of providing enough additional semantic information to make a document
more accessible by facilitating better visual rendering, voice rendering, or specialized processing, controlling the exact
notation used becomes more of an issue.

MathML elements accept an attribute
other (see
Section 7.2.3 Attributes for unspecified data), which can be used to specify things not specifically documented in MathML. On content tags, this attribute can be used
by an author to express a
preference between equivalent forms for a particular content element construct, where the selection of the presentation has nothing
to do with the semantics. Examples might be

inline or displayed equations

script-style fractions

use of
x with a dot for a derivative over dx/dt

Thus, if a particular renderer recognized a display attribute to select between script-style and display-style fractions,
an author might write

The information provided in the
other attribute is intended for use by specific renderers or processors, and therefore, the permitted values are determined by
the renderer being used. It is legal for a renderer to ignore this information. This might be intentional, as in the case
of a publisher imposing a house style, or simply because the renderer does not understand them, or is unable to carry them
out.

4.4 The Content Markup Elements

This section provides detailed descriptions of the MathML content tags. They are grouped in categories that broadly reflect
the area of mathematics from which they come, and also the grouping in the MathML DTD. There is no linguistic difference in
MathML between operators and functions. Their separation here and in the DTD is for reasons of historical usage.

When working with the content elements, it can be useful to keep in mind the following.

The role of the content elements is analogous to data entry in a mathematical system. The information that is provided is
there to facilitate the successful parsing of an expression as the intended mathematical object by a receiving application.

MathML content elements do not by themselves
"perform" any mathematical evaluations or operations. They do not
"evaluate" in a browser and any
"action" that is ultimately taken on those objects is determined entirely by the receiving mathematical application. For example,
editing programs and applications geared to computation for the lower grades would typically leave 3 + 4 as is, whereas computational
systems targeting a more advanced audience might evaluate this as 7. Similarly, some computational systems might evaluate
sin(0) to 0, whereas others would leave it unevaluated. Yet other computational systems might be unable to deal with pure
symbolic expressions like sin(x) and may even regard them as data entry errors. None of this has any bearing on the correctness of the original MathML representation.
Where evaluation is mentioned at all in the descriptions below, it is merely to help clarify the meaning of the underlying
operation.

Apart from the instances where there is an explicit interaction with presentation tagging, there is no required rendering
(visual or aural) - only a suggested default. As such, the presentations that are included in this section are merely to
help communicate to the reader the intended mathematical meaning by association with the same expression written in a more
traditional notation.

4.4.1 Token Elements

4.4.1.1 Number (cn)

4.4.1.1.1 Discussion

The cn element is used to specify actual
numerical constants. The content model must provide sufficient information
that a number may be entered as data into a computational system. By
default, it represents a signed real number in base 10. Thus, the content
normally consists of PCDATA restricted to a sign, a string of
decimal digits and possibly a decimal point, or alternatively one of the
predefined symbolic constants such as &pi;.

The cn element uses the attribute type to represent other types of numbers such as, for
example, integer, rational, real or complex, and uses the attribute base to specify the numerical base.

In addition to simple PCDATA, cn
accepts as content PCDATA separated by the (empty) element sep. This determines the different parts needed to
construct a rational or complex-cartesian number.

The cn element may also contain arbitrary
presentation markup in its content (see Chapter 3 Presentation Markup) so that its
presentation can be very elaborate.

Alternative input notations for numbers are possible, but must be
explicitly defined by using the definitionURL and
encoding attributes, to refer to a written
specification of how a sequence of real numbers separated by <sep/> should be interpreted.

4.4.1.1.4 Default Rendering

By default, a contiguous block of
PCDATA contained in a
cn element should render as if it were wrapped in an
mn presentation element.

If an application supports bidirectional text rendering, then the
rendering within a cn element follows the Unicode
bidirectional rendering rules just as if it were wrapped in an
mn presentation element.

Similarly, presentation markup contained in a
cn element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were wrapped in an
mrow element, with contiguous blocks of
PCDATA wrapped in
mn elements.

However, not all mathematical systems that encounter content based tagging do visual or aural rendering. The receiving applications
are free to make use of a number in the manner in which they normally handle numerical data. Some systems might simplify the
rational number 12342/2342342 to 6171/1171171 while pure floating point based systems might approximate this as 0.5269085385e-2.
All numbers might be re-expressed in base 10. The role of MathML is simply to record enough information about the mathematical
object and its structure so that it may be properly parsed.

The following renderings of the above MathML expressions are included both to help clarify the meaning of the corresponding
MathML encoding and as suggestions for authors of rendering applications. In each case, no mathematical evaluation is intended
or implied.

12345.7

12345

AB316

12342 / 2342342

12.3 + 5 i

Polar( 2 , 3.1415 )

4.4.1.2 Identifier (ci)

4.4.1.2.1 Discussion

The
ci element is used to name an identifier in a MathML expression (for example a variable). Such names are used to identify mathematical
objects. By default they are assumed to represent complex scalars. The
ci element may contain arbitrary presentation markup in its content (see
Chapter 3 Presentation Markup) so that its presentation as a symbol can be very elaborate.

The
ci element uses the
type attribute to specify the basic type of object that it represents. While any CDATA string
is a valid type, the predefined types include
"integer",
"rational",
"real",
,
"complex",
"complex-polar",
"complex-cartesian",
"constant", "function" and more generally, any of the names of the MathML container elements (e.g.
vector) or their type values.
For a more advanced treatment of types, the type attribute is inappropriate. Advanced types require
significant structure of their own (for example, vector(complex)) and are probably best constructed as
mathematical objects and then associated with a MathML expression through use of the semantics element. Additional
information on this topic is planned. See the MathML website for more information.

The definitionURL attribute can be used to associate additional properties with a ci element.
See the discussion of bound variables (Section 4.4.5.6 Bound variable (bvar)) for a discussion of an important instance of this.

4.4.1.2.2 Examples

<ci> x </ci>

<ci type="vector"> V </ci>

<ci>
<msub>
<mi>x</mi>
<mi>a</mi>
</msub>
</ci>

4.4.1.2.3 Default Rendering

If the content of a
ci element is tagged using presentation tags, that presentation is used. If no such tagging is supplied then the
PCDATA content is rendered as if it were the content of an
mi element.

If an application supports bidirectional text rendering, then the
rendering within a ci element follows the Unicode
bidirectional rendering rules just as if it were wrapped in an
mi presentation element.

A renderer may wish to make use of the value of the type attribute to improve on this. For example, a symbol of type
vector might be rendered using a bold face. Typical renderings of the above symbols are:

4.4.1.3 Externally defined symbol (csymbol)

4.4.1.3.1 Discussion

The
csymbol element allows a writer to create an element in MathML whose semantics are externally defined (i.e. not in the core MathML
content). The element can then be used in a MathML expression as for example an operator or constant. Attributes are used
to give the syntax and location of the external definition of the symbol semantics.

Use of
csymbol for referencing external semantics can be contrasted with use of the
semantics to attach additional information in-line (i.e. within the MathML fragment) to a MathML construct. See
Section 4.2.6 Syntax and Semantics.

4.4.1.3.2 Attributes

All attributes are
CDATA:

definitionURL

Pointer to external definition of the semantics of the symbol. MathML does not specify a particular syntax in which this definition
should be written.

encoding

Gives the syntax of the definition pointed to by definitionURL. An application can then test the value of this attribute to
determine whether it is able to process the target of the
definitionURL. This syntax might be text, or a formal syntax such as OpenMath.

4.4.1.3.4 Default Rendering

By default, a contiguous block of
PCDATA contained in a
csymbol element should render as if it were wrapped in an
mo presentation element.

If an application supports bidirectional text rendering, then the
rendering within a csymbol element follows the Unicode
bidirectional rendering rules just as if it were wrapped in an
mo presentation element.

Similarly, presentation markup contained in a
csymbol element should render as it normally would. A mixture of
PCDATA and presentation markup should render as if it were contained wrapped in an
mrow element, with contiguous blocks of
PCDATA wrapped in
mo elements. The examples above would render by default as

As
csymbol is used to support reference to externally defined semantics, it is a MathML error to have embedded content MathML elements
within the
csymbol element.

4.4.2 Basic Content Elements

4.4.2.1 Apply (apply)

4.4.2.1.1 Discussion

The
apply element allows a function or operator to be applied to its arguments. Nearly all expression construction in MathML content
markup is carried out by applying operators or functions to arguments. The first child of
apply is the operator to be applied, with the other child elements as arguments or qualifiers.

The
apply element is conceptually necessary in order to distinguish between a function or operator, and an instance of its use. The
expression constructed by applying a function to 0 or more arguments is always an element from the codomain of the function.

Proper usage depends on the operator that is being applied. For example, the
plus operator may have zero or more arguments, while the
minus operator requires one or two arguments to be properly formed.

If the object being applied as a function is not already one of the elements known to be a function (such as
fn (deprecated),
sin or
plus) then it is treated as if it were the content of an
fn element.

Some operators such as user defined functions defined using the declare
or csymbol elements,
diff and
int make use of
"named" arguments. These special arguments are elements that appear as children of the
apply element and identify
"parameters" such as the variable of differentiation or the domain of integration. These elements are discussed further in
Section 4.2.3.2 Operators taking Qualifiers.

4.4.2.1.2 Examples

<apply>
<factorial/>
<cn>3</cn>
</apply>

<apply>
<plus/>
<cn>3</cn>
<cn>4</cn>
</apply>

<apply>
<sin/>
<ci>x</ci>
</apply>

4.4.2.1.3 Default Rendering

A mathematical system that has been passed an
apply element is free to do with it whatever it normally does with such mathematical data. It may be that no rendering is involved
(e.g. a syntax validator), or that the
"function application" is evaluated and that only the result is rendered (e.g. sin(0)
0).

When an unevaluated
"function application" is rendered there are a wide variety of appropriate renderings. The choice often depends on the function
or operator being applied. Applications of basic operations such as
plus are generally presented using an infix notation while applications of
sin would use a more traditional functional notation such as sin(x). Consult the default rendering for the operator being applied.

Applications of user-defined functions (see
csymbol,
fn) that are not evaluated by the receiving or rendering application would typically render using a traditional functional notation
unless an alternative presentation is specified using the
semantics tag.

4.4.2.2 Relation (reln)

4.4.2.2.1 Discussion

The
reln element was used in MathML 1.0 to construct an equation or relation. Relations were constructed in a manner exactly analogous
to the use of
apply. This usage is deprecated in MathML 2.0 in favor of the more generally usable
apply.

The first child of
reln is the relational operator to be applied, with the other child elements acting as arguments. See
Section 4.2.4 Relations for further details.

4.4.2.2.2 Examples

<reln>
<eq/>
<ci> a </ci>
<ci> b </ci>
</reln>

<reln>
<lt/>
<ci> a </ci>
<ci> b </ci>
</reln>

4.4.2.2.3 Default Rendering

4.4.2.3 Function (fn)

4.4.2.3.1 Discussion

The
fn element makes explicit the fact that a more general (possibly constructed) MathML object is being used in the same manner
as if it were a pre-defined function such as
sin or
plus.

fn has exactly one child element, used to give the name (or presentation form) of the function. When
fn is used as the first child of an apply, the number of following arguments is determined by the contents of the
fn.

In MathML 1.0,
fn was also the primary mechanism used to extend the collection of
"known" mathematical functions. The fn element has been deprecated. To extend the collection
of known mathematical functions without using the fn element, use the more generally applicable
csymbol element or use a declare in conjunction with a
lambda expression.

4.4.2.3.2 Examples

4.4.2.3.3 Default Rendering

An
fn object is rendered in the same way as its content. A rendering application may add additional adornments such as parentheses
to clarify the meaning.

4.4.2.4 Interval (interval)

4.4.2.4.1 Discussion

The
interval element is used to represent simple mathematical intervals of the real number line. It takes an attribute
closure, which can take on any of the values
"open",
"closed",
"open-closed", or
"closed-open", with a default value of
"closed".

When used in the body of an apply element it serves as a shorthand notation for
a domainofapplication.

4.4.2.4.2 Examples

4.4.2.4.3 Default Rendering

4.4.2.5 Inverse (inverse)

4.4.2.5.1 Discussion

The
inverse element is applied to a function in order to construct a generic expression for the functional inverse of that function.
(See also the discussion of
inverse in
Section 4.2.1.5 The inverse construct). As with other MathML functions,
inverse may either be applied to arguments, or it may appear alone, in which case it represents an abstract inversion operator acting
on other functions.

A typical use of the
inverse element is in an HTML document discussing a number of alternative definitions for a particular function so that there is
a need to write and define
f(-1)(x). To associate a particular definition with
f(-1), use the
definitionURL and
encoding attributes.

4.4.2.5.3 Default Rendering

The default rendering for a functional inverse makes use of a
parenthesized exponent as in f(-1)(x).

4.4.2.6 Separator (sep)

4.4.2.6.1 Discussion

The sep element is used to separate PCDATA
into separate tokens for parsing the contents of the various specialized
forms of the cn elements. For example, sep is used when specifying the real and imaginary
parts of a complex number (see Section 4.4.1 Token Elements). If it
occurs between MathML elements, it is a MathML error.

4.4.2.6.2 Examples

<cn type="complex-cartesian"> 3 <sep/> 4 </cn>

4.4.2.6.3 Default Rendering

4.4.2.7 Condition (condition)

4.4.2.7.1 Discussion

The condition element is used to assert
that a Boolean valued expression should be true.
The conditions may be specified in terms of relations that are
to be satisfied ,
including general relationships such as set membership.
When used in conjunction with the bound variables of
an apply element, it serves as a shorthand notation for
the domainofapplication defined by having
n-tuples of values of the bound variables of the surrounding apply element
included in the domain when the conditions placed on them
in this way are satisfied and excluded otherwise.

It is used to define general sets and lists in situations where the
elements cannot be explicitly enumerated. Condition contains either a
single apply or reln element (deprecated);
the apply element
is used to construct compound conditions. For example, it is used below to
describe the set of all x such that x < 5. See the
discussion on sets in Section 4.4.6 Theory of Sets. See Section 4.2.5 Conditions for further details.

4.4.2.7.3 Default Rendering

4.4.2.8 Declare (declare)

4.4.2.8.1 Discussion

The
declare construct has two primary roles. The first is to change or set the default attribute values for a specific mathematical object.
The second is to establish an association between a
"name" and an object. Once a declaration is in effect, the
"name" object acquires the new attribute settings, and (if the second object is present) all the properties of the associated
object.

The various attributes of the
declare element assign properties to the object being declared or determine where the declaration is in effect.
The list of allowed attributes varies depending on the object involved as it always includes the
attributes associated with that object.

The scope of a declaration is "local" to the surrounding container element.
The scope attribute can only be assigned to
"local", but is intended to support future extensions.
As discussed in
Section 4.3.2.8
scope, MathML contains no provision for making document-wide declarations at present, though it is anticipated that this capability
will be added in future revisions of MathML, when supporting technologies become available.

declare takes one or two children. The first child, which
is mandatory, is the object affected by the declaration.
This is usually a ci element
providing the identifier that is being declared as in:

<declare type="vector"> <ci> V </ci> </declare>

The second child, which is optional, is a constructor initializing the variable:

The constructor type and the type of the element declared must agree. For example, if the type attribute of the declaration
is
function, the second child (constructor) must be an element that can serve as a function.
(This would typically be something like a
csymbol element, a ci element,
a lambda element, or any of the defined functions in the basic set of content tags.) If no type is specified in the declaration then
the type attribute of the declared name is set to the type of the constructor (second child) of the declaration.

An important case is when the first child is an identifier, and the second child is a semantics tag enclosing that identifier.
In this case all uses of the identifier acquire the associations implied by the use of the semantics element.
without having to write out the full semantics element for every use.

The actual instances of a declared ci element are normally recognized
by comparing their content with that of the declared element.
Equality of two elements is determined by comparing the XML information set
of the two expressions after XML space normalization.

When the content is more complex, semantics elements are involved, or the author
simply wants to use multiple presentations for emphasis
without losing track of the relationship to the declared
instance the author may choose to make the correspondence explicit by placing
an id attribute on a declared instance and referring back to it using a
definitionURL attribute on the matching instances of the ci element
as in the following example.

associates the name
J with a one-variable function defined so that
J(y) = ln
y. (Note that because of the type attribute of the
declare element, the second argument must be something of function type
, namely a known function like
sin, or a
lambda construct.)

The
type attribute on the declaration is only necessary if the type cannot be inferred from the type of the second argument.

Even when a declaration is in effect it is still possible to override attributes values selectively as in
<ci type="set"> S
</ci>. This capability is needed in order to write statements of the form
"Let
s be a member of
S".

4.4.2.8.4 Default Rendering

Since the
declare construct is not directly rendered, most declarations are likely to be invisible to a reader. However, declarations can produce
quite different effects in an application which evaluates or manipulates MathML content. While the declaration

is active the symbol v acquires all the properties of the vector,
and even its dimension and components have meaningful values. This may
affect how v is rendered by some applications, as well as how it
is treated mathematically.

4.4.2.9 Lambda (lambda)

4.4.2.9.1 Discussion

The lambda element is used to construct a user-defined function from
an expression, bound variables, and
qualifiers. In a lambda construct with n
(possibly 0) bound variables, the first n children are bvar
elements that identify the variables that are used as placeholders in the last
child for actual parameter values. The bound variables
can be restricted by an optional domainofapplication qualifier. The
meaning of the lambda construct is an n-ary function that
returns the expression in the last child where the bound variables are replaced
with the respective arguments.
See Section 4.2.2.2 Constructors for further details.

Such constructs are often used in conjunction with
declare to construct new functions.

The domainofapplication child restricts the possible
values of the arguments of the constructed function. For instance, the
following two lambda constructs are representations of a function on
the integers.

If a lambda construct does not contain bound variables, then the
arity of the constructed function is unchanged, and the lambda
construct is redundant, unless it also contains a
domainofapplication construct that restricts existing functional
arguments, as in this example, which is a variant representation for the
function above.

In particular, if the last child of a lambda construct is not a
function, say a number, then the lambda construct will not be a
function, but the same number. Of course, in this case a
domainofapplication does not make sense

4.4.2.9.3 Default Rendering

4.4.2.10 Function composition (compose)

4.4.2.10.1 Discussion

The compose element represents the function
composition operator. Note that MathML makes no assumption about the domain
and codomain of the constituent functions in a composition; the domain of the
resulting composition may be empty.

To override the default semantics for the compose element, or to associate a more specific
definition for function composition, use the definitionURL and encoding attributes. See Section 4.2.3 Functions, Operators and Qualifiers
for further details.

4.4.2.10.3 Default Rendering

4.4.2.11 Identity function (ident)

4.4.2.11.1 Discussion

The ident element represents the identity
function. MathML makes no assumption about the function space in which the
identity function resides. That is, proper interpretation of the domain
(and hence codomain) of the identity function depends on the context in which
it is used.

4.4.2.12.2 Examples

4.4.2.12.3 Default Rendering

4.4.2.13 codomain (codomain)

4.4.2.13.1 Discussion

The codomain element denotes the codomain of a given function, which is a set
containing all values taken by the function. It is not necessarily the case that every point in
the codomain is generated by the function applied to some point of the domain. (For example I may know
that a function is integer-valued, so its codomain is the integers, without knowing (or stating) which
subset of the integers is mapped to by the function.)

4.4.2.14.2 Examples

4.4.2.14.3 Default Rendering

4.4.2.15 Domain of Application (domainofapplication)

4.4.2.15.1 Discussion

The domainofapplication element is a qualifier which denotes the domain over which a given function
is being applied. It is intended to be a more general alternative to specification of this
domain using such qualifier elements as bvar, lowlimit
or condition.

4.4.2.15.2 Examples

4.4.2.15.3 Default Rendering

The default rendering depends on the particular function being applied.

4.4.2.16 Piecewise declaration
(piecewise, piece,
otherwise)

4.4.2.16.1 Discussion

The
piecewise,
piece, and
otherwise
elements are used to support "piecewise" declarations of the form "
H(x) = 0 if x less than 0,
H(x) = 1 otherwise".

The declaration is constructed using the piecewise element.
This contains zero or more piece elements, and optionally
one otherwise element. Each piece
element contains exactly two children. The first child defines the value taken by the piecewise
expression when the condition specified in the associated second child of the piece is true.
The degenerate case of no piece elements and no otherwise element is treated as
undefined for all values of the domain.

otherwise allows the specification of a value to be taken by the
piecewise function when none of the conditions (second child elements of the
piece elements) is true, i.e. a default value.

It should be noted that no "order of execution" is implied by the ordering of the piece
child elements within piecewise. It is the responsibility of the author
to ensure that the subsets of the function domain defined by the second children of the piece elements are disjoint,
or that, where they overlap, the values of the corresponding first children of the piece
elements coincide. If this is not the case, the meaning of the expression is undefined.

4.4.2.16.3 Default Rendering

4.4.3 Arithmetic, Algebra and Logic

4.4.3.1 Quotient (quotient)

4.4.3.1.1 Discussion

The quotient element is the operator used for
division modulo a particular base. When the quotient operator is applied to integer arguments
a and b, the result is the "quotient of
a divided by b". That is, quotient returns the unique integer q such
that a = qb + r. (In common usage,
q is called the quotient and r is the remainder.)

The quotient element takes the attribute definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.3.1.2 Example

<apply>
<quotient/>
<ci> a </ci>
<ci> b </ci>
</apply>

Various mathematical applications will use this data in different ways. Editing applications might choose an image such as
shown below, while a computationally based application would evaluate it to 2 when
a=13 and
b=5.

4.4.3.1.3 Default Rendering

There is no commonly used notation for this concept. Some possible renderings are

quotient of
a divided by
b

integer part of
a / b

4.4.3.2 Factorial (factorial)

4.4.3.2.1 Discussion

The factorial element is used to construct factorials.

The factorial element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.3.3.2 Example

<apply>
<divide/>
<ci> a </ci>
<ci> b </ci>
</apply>

As a MathML expression, this does not evaluate. However, on receiving such an expression, some applications may attempt to
evaluate and simplify the value. For example, when
a=5 and
b=2 some mathematical applications may evaluate this to 2.5 while others will treat is as a rational number.

4.4.3.3.3 Default Rendering

4.4.3.4 Maximum and minimum (max,
min)

4.4.3.4.1 Discussion

The elements
max and
min are used to compare the values of their arguments. They return the maximum and minimum of these values respectively.

The
max and
min elements take the
definitionURL and
encoding attributes that can be used to override the default semantics.

Note that the bound variable must be stated even if it might
be implicit in conventional notation. In MathML1.0, the bound variable
and expression to be evaluated (x) could be omitted in the
example below: this usage is deprecated in MathML2.0 in favor of
explicitly stating the bound variable and expression in all cases:

4.4.3.4.3 Default Rendering

4.4.3.5 Subtraction (minus)

4.4.3.5.1 Discussion

The minus element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

The minus element can be used as a unary
arithmetic operator (e.g. to represent - x), or as a
binary arithmetic operator (e.g. to represent x-
y).

4.4.3.5.2 Example

<apply> <minus/>
<ci> x </ci>
<ci> y </ci>
</apply>

If this were evaluated at
x=5 and
y=2 it would yield 3.

4.4.3.5.3 Default Rendering

4.4.3.6 Addition (plus)

4.4.3.6.1 Discussion

The plus element is the addition operator.

The plus element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

The plus element is an n-ary arithmetic
operator (see Section 4.2.3 Functions, Operators and Qualifiers).
The operands are usually listed explicitly.
As an n-ary operator, these operands may in principle be generated by allowing
a function or expression to vary over a domain of application. However, such expressions
can be represented explicitly by using Section 4.4.7.1 Sum (sum) so the plus does not normally
take qualifiers.

4.4.3.6.2 Example

<apply>
<plus/>
<ci> x </ci>
<ci> y </ci>
<ci> z </ci>
</apply>

If this were evaluated at
x = 5,
y = 2 and
z = 1 it would yield 8.

4.4.3.6.3 Default Rendering

4.4.3.7 Exponentiation (power)

4.4.3.7.1 Discussion

The power element is a generic exponentiation
operator. That is, when applied to arguments a and b, it
returns the value of "a to the power of
b".

The
power element takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

4.4.3.7.2 Example

<apply>
<power/>
<ci> x </ci>
<cn> 3 </cn>
</apply>

If this were evaluated at
x= 5 it would yield 125.

4.4.3.7.3 Default Rendering

4.4.3.8 Remainder (rem)

4.4.3.8.1 Discussion

The rem element is the operator that returns the
"remainder" of a division modulo a particular base. When the
rem operator is applied to integer arguments
a and b, the result is the "remainder of
a divided by b". That is, rem returns the unique integer, r such that
a = qb+ r, where r <
q. (In common usage, q is called the quotient and
r is the remainder.)

The
rem element takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

4.4.3.8.2 Example

<apply>
<rem/>
<ci> a </ci>
<ci> b </ci>
</apply>

If this were evaluated at
a = 15 and
b = 8 it would yield 7.

4.4.3.8.3 Default Rendering

4.4.3.9 Multiplication (times)

4.4.3.9.1 Discussion

The
times element is the multiplication operator.
The operands are usually listed explicitly.
As an n-ary operator, these operands may in principle be generated by allowing
a function or expression to vary over a domain of application. However, such expressions
can be represented explicitly by using Section 4.4.7.2 Product (product) so the times does not normally
take qualifiers.

times takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

4.4.3.9.2 Example

<apply>
<times/>
<ci> a </ci>
<ci> b </ci>
</apply>

If this were evaluated at
a = 5.5 and
b = 3 it would yield 16.5.

4.4.3.9.3 Default Rendering

4.4.3.10 Root (root)

4.4.3.10.1 Discussion

The root element is used to construct roots. The
kind of root to be taken is specified by a degree element, which should be given as the second child
of the apply element enclosing the root element. Thus, square roots correspond to the case
where degree contains the value 2, cube roots
correspond to 3, and so on. If no degree is
present, a default value of 2 is used.

The root element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.3.10.2 Example

4.4.3.10.3 Default Rendering

4.4.3.11 Greatest common divisor (gcd)

4.4.3.11.1 Discussion

The gcd element is used to denote the greatest
common divisor of its arguments.

The gcd takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

The gcd element is an
n-ary operator (see
Section 4.2.3 Functions, Operators and Qualifiers).
As an n-ary operator, its operands may also be generated by allowing
a function or expression to vary over a domain of application. Therefore it may take qualifiers.

4.4.3.11.2 Example

<apply> <gcd/>
<ci> a </ci>
<ci> b </ci>
<ci> c </ci>
</apply>

If this were evaluated at
a = 15,
b = 21,
c = 48, it would yield 3.

4.4.3.11.3 Default Rendering

This default rendering is English-language locale specific: other locales
may have different default renderings.

4.4.3.12 And (and)

4.4.3.12.1 Discussion

The and element is the Boolean
"and" operator.

The
and element takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

4.4.3.16.2 Example

Mathematical applications designed for the evaluation of such expressions would evaluate this to
"true" when
a =
"false" and
b =
"true".

4.4.3.16.3 Default Rendering

4.4.3.17 Universal quantifier (forall)

4.4.3.17.1 Discussion

The
forall element represents the universal quantifier of logic. It must be used in conjunction with one or more bound variables, an
optional
condition element, and an assertion. In MathML 1.0, the
reln element was also permitted here: this usage is now deprecated.

The
forall element takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

4.4.3.17.3 Default Rendering

Note:

The second and third examples in this section are correct MathML expressions of False mathematical statements.

4.4.3.18 Existential quantifier (exists)

4.4.3.18.1 Discussion

The exists element represents the existential
quantifier of logic. It must be used in conjunction with one or more bound
variables, an optional condition element, and an
assertion, which may take the form of either an apply or reln element.

The exists element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.3.20.2 Example

4.4.3.20.3 Default Rendering

4.4.3.21 Argument (arg)

4.4.3.21.1 Discussion

The arg operator (introduced in MathML 2.0)
gives the "argument" of a complex number, which is the angle
(in radians) it makes with the positive real axis. Real negative numbers
have argument equal to + .

The arg element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.3.23.2 Example

A MathML-aware evaluation system would return the
y component, suitably encoded.

4.4.3.23.3 Default Rendering

4.4.3.24 Lowest common multiple (lcm)

4.4.3.24.1 Discussion

The
lcm element (introduced in MathML 2.0) is used to denote the lowest common
multiple of its arguments.

The
lcm takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

The
lcm element is an
n-ary operator (see
Section 4.2.3 Functions, Operators and Qualifiers).
As an n-ary operator, its operands may also be generated by allowing
a function or expression to vary over a domain of application. Therefore it may take qualifiers.

4.4.3.24.2 Example

<apply> <lcm/>
<ci> a </ci>
<ci> b </ci>
<ci> c </ci>
</apply>

If this were evaluated at
a = 2,
b = 4,
c = 6 it would yield 12.

4.4.3.24.3 Default Rendering

This default rendering is English-language locale specific: other locales
may have different default renderings.

4.4.3.25 Floor (floor)

4.4.3.25.1 Discussion

The
floor element (introduced in MathML 2.0) is used to denote the
round-down (towards -infinity) operator.

The
floor takes the
definitionURL and
encoding attributes, which can be used to override the default semantics.

4.4.4.9.2 Example

<apply>
<factorof/>
<ci> a </ci>
<ci> b </ci>
</apply>

4.4.4.9.3 Default Rendering

4.4.5 Calculus and Vector Calculus

4.4.5.1 Integral (int)

4.4.5.1.1 Discussion

The int element is the operator element for an
integral. Optional bound variables serve as the integration
variables and definite integrals are indicated by providing a domain of integration.
This may be provided by an optional domainofapplication element or one of the
shortcut representations of the domain of application (see Section 4.2.3.2 Operators taking Qualifiers).
For example, the integration variable and domain of application
can be given by the child elements lowlimit, uplimit and bvar in the
enclosing apply element. The integrand is also
specified as a child element of the enclosing apply
element.

The int element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.5.1.3 Default Rendering

4.4.5.2 Differentiation (diff)

4.4.5.2.1 Discussion

The diff element is the differentiation operator
element for functions of a single variable. It may be applied directly to
an actual function such as sine or cosine, thereby denoting a function which is
the derivative of the original function, or it can be applied to an expression
involving a single variable such as sin(x), or cos(x). or a
polynomial in x. For the expression case the actual variable is
designated by a bvar element that is a child of the
containing apply element. The bvar element may also contain a degree element, which specifies the order of the
derivative to be taken.

The diff element takes the definitionURL and encoding
attributes, which can be used to override the
default semantics.

4.4.5.2.3 Default Rendering

4.4.5.3 Partial Differentiation (partialdiff)

4.4.5.3.1 Discussion

The partialdiff element is the partial
differentiation operator element for functions or algebraic expressions in several
variables.

In the case of algebraic expressions, the bound variables are given by bvar
elements, which are children of the containing apply element. The bvar elements
may also contain degree element, which specify
the order of the partial derivative to be taken in that variable.

For the expression case the actual variable is
designated by a bvar element that is a child of the
containing apply element. The bvar elements may also contain a degree element, which specifies the order of the
derivative to be taken.

Where a total degree of differentiation must be specified, this is indicated by use of a
degree element at the top level, i.e. without any associated
bvar, as a child
of the containing apply element.

For the case of partial differentiation of a function, the containing apply takes
two child elements: firstly a list of indices indicating by position
which coordinates are involved in
constructing the partial derivatives, and secondly the actual function to be partially differentiated.
The coordinates may be repeated.

The partialdiff element takes the definitionURL and encoding
attributes, which can be used to override the
default semantics.

4.4.5.3.3 Default Rendering

4.4.5.4 Lower limit (lowlimit)

4.4.5.4.1 Discussion

The lowlimit element is the container element
used to indicate the "lower limit" of an operator using
qualifiers. For example, in an integral, it can be used to specify the
lower limit of integration. Similarly, it can be used to specify the lower
limit of an index for a sum or product.

The meaning of the lowlimit element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 Operators taking Qualifiers.

4.4.5.4.3 Default Rendering

The default rendering of the
lowlimit element and its contents depends on the context. In the preceding example, it should be rendered as a subscript to the integral
sign:

Consult the descriptions of individual operators that make use of the
lowlimit construct for default renderings.

4.4.5.5 Upper limit (uplimit)

4.4.5.5.1 Discussion

The uplimit element is the container element
used to indicate the "upper limit" of an operator using
qualifiers. For example, in an integral, it can be used to specify the
upper limit of integration. Similarly, it can be used to specify the upper
limit of an index for a sum or product.

The meaning of the uplimit element depends on
the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 Operators taking Qualifiers.

4.4.5.5.3 Default Rendering

The default rendering of the uplimit element and
its contents depends on the context. In the preceding example, it should be
rendered as a superscript to the integral sign:

Consult the descriptions of individual operators that make use of the
uplimit construct for default renderings.

4.4.5.6 Bound variable (bvar)

4.4.5.6.1 Discussion

The bvar element is the container element for
the "bound variable" of an operation. For example, in an
integral it specifies the variable of integration. In a derivative, it
indicates the variable with respect to which a function is being
differentiated. When the bvar element is used to
qualify a derivative, it may contain
a child degree element that specifies the order of
the derivative with respect to that variable. The bvar element is also used for the internal variable in
a number of operators taking qualifiers, including user defined operators,
sums and products and for the bound variable used with the universal and
existential quantifiers forall and exists.
When a bvar element has more than one
child element, the elements may appear in any order.

Instances of the
bound variables are normally recognized by comparing the XML information
sets of the relevant ci elements after first carrying out XML space
normalization. Such identification can be made explicit by placing an
id on the ci element in the bvar element and
referring to it using the definitionURL attribute on all other
instances. An example of this approach is

This id based approach is especially helpful when constructions
involving bound variables are nested.

It can be necessary to associate additional
information with a bound variable or one or more instances of it.
The information might be something like a detailed mathematical type or an alternative presentation or encoding.
Such associations are accomplished in the standard way
by replacing a ci element (even inside the bvar element) by a semantics element containing both it and the additional information.
Recognition of and instance of the bound variable is still based on the actual ci elements and not the semantics elements or anything else
they may contain. The id based approach outlined above may still
be used.

The meaning of the bvar element depends on the
context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking
qualifiers, consult Section 4.2.3.2 Operators taking Qualifiers.

4.4.5.6.3 Default Rendering

The default rendering of the
bvar element and its contents depends on the context. In the preceding examples, it should be rendered as the
x in the dx of the integral, and as the
x in the denominator of the derivative symbol, respectively:

Note that in the case of the derivative, the default rendering of the
degree child of the
bvar element is as an exponent.

Consult the descriptions of individual operators that make use of the
bvar construct for default renderings.

4.4.5.7 Degree (degree)

4.4.5.7.1 Discussion

The degree element is the container element for
the "degree" or "order" of an operation. There
are a number of basic mathematical constructs that come in families, such as
derivatives and moments. Rather than introduce special elements for each of
these families, MathML uses a single general construct, the degree element for this concept of
"order".

The meaning of the
degree element depends on the context it is being used in. For further details about how
qualifiers are used in conjunction with operators taking qualifiers, consult
Section 4.2.3.2 Operators taking Qualifiers.

4.4.5.11.2 Example

4.4.5.11.3 Default Rendering

4.4.6 Theory of Sets

4.4.6.1 Set (set)

4.4.6.1.1 Discussion

The
set element is the container element that constructs a set of elements. The elements of a set can be defined
either by explicitly listing the elements, or by evaluating a function over a domain of application
as described in Section 4.2.3.2 Operators taking Qualifiers.

This constructs the set of all natural numbers less than 5, i.e. the set {0, 1, 2, 3, 4}.
In general a set can be constructed by providing a function and a domain of application. The elements of the
set correspond to the values obtained by evaluating the function at the points of the domain.
The qualifications defined by a domainofapplication element can also be abbreviated
in several ways including just a condition element placing constraints directly on the bound variables
as in this example

4.4.6.1.3 Default Rendering

4.4.6.2 List (list)

4.4.6.2.1 Discussion

The
list element is the container element that constructs a list of elements. Elements can be defined either by
explicitly listing the elements, or by evaluating a function over a domain of application
as described in Section 4.2.3.2 Operators taking Qualifiers.

Lists differ from sets in that there is an explicit order to the elements. Two orders are supported: lexicographic and numeric.
The kind of ordering that should be used is specified by the
order attribute.

4.4.6.2.3 Default Rendering

4.4.6.3 Union (union)

4.4.6.3.1 Discussion

The union element is the operator for a
set-theoretic union or join of sets.
The operands are usually listed explicitly.

The union element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

The union element is an n-ary set
operator (see Section 4.2.3 Functions, Operators and Qualifiers).
As an n-ary operator, its operands may also be generated by allowing
a function or expression to vary over a domain of application. Therefore it may take qualifiers.

4.4.6.3.3 Default Rendering

4.4.6.4 Intersect (intersect)

4.4.6.4.1 Discussion

The intersect element is the operator for the
set-theoretic intersection or meet of sets.
The operands are usually listed explicitly.

The intersect element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

The intersect element is an n-ary set
operator (see Section 4.2.3 Functions, Operators and Qualifiers).
As an n-ary operator, its operands may also be generated by allowing
a function or expression to vary over a domain of application. Therefore it may take qualifiers.

4.4.6.12.2 Example

4.4.6.12.3 Default Rendering

4.4.6.13 Cartesian product (cartesianproduct)

4.4.6.13.1 Discussion

The cartesianproduct element is the operator element for
the Cartesian product of two or more sets. If A and B are two sets, then
the Cartesian product of A and B is the set of all pairs (a,b)
with a in A and b in B.

The cartesianproduct element takes the attributes definitionURL and encoding that can be used to override the
default semantics.

4.4.6.13.2 Example

4.4.6.13.3 Default Rendering

4.4.7 Sequences and Series

4.4.7.1 Sum (sum)

4.4.7.1.1 Discussion

The sum element denotes the summation
operator.
The most general form of a sum specifies the terms of the sum by using a domainofapplication
element. If no bound variables are specified then terms of the sum correspond to those produced by
evaluating the function that is provided at the points of the domain, while if
bound variables are present they are the index of summation and they take
on the values of points in the domain. In this case the terms of the sum correspond to the values of the
expression that is provided, evaluated at those points. Depending on the structure of the domain,
the domain of summation can be abbreviated by using uplimit and lowlimit
to specify upper and lower limits for the sum.

The sum element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.7.1.3 Default Rendering

4.4.7.2 Product (product)

4.4.7.2.1 Discussion

The product element denotes the product
operator. Upper and lower limits for the index of a product can be specified using
uplimit and lowlimit. More general
domains for the indices can be specified using a condition
involving the bound variables or a domainofapplication element.
The index for the product is specified by a
bvar element.

The product element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.7.2.3 Default Rendering

4.4.7.3 Limit (limit)

4.4.7.3.1 Discussion

The limit element represents the operation of
taking a limit of a sequence. The limit point is expressed by specifying a
lowlimit and a bvar, or by
specifying a condition on one or more bound
variables.

The limit element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.7.3.3 Default Rendering

4.4.7.4 Tends To (tendsto)

4.4.7.4.1 Discussion

The tendsto element is used to express the
relation that a quantity is tending to a specified value. While this is used primarily as part
of the statement of a mathematical limit, it exists as a construct on its own to allow one to capture mathematical
statements such as "As x tends to y," and to provide a building block to construct more general kinds of limits that
are not explicitly covered by the recommendation.

The tendsto element takes the attributes type to set the direction from which the limiting
value is approached.

4.4.8.3.2 Example

<apply>
<ln/>
<ci> a </ci>
</apply>

If
a =
e, (where e is the base of the natural logarithms) this will yield the value 1.

4.4.8.3.3 Default Rendering

4.4.8.4 Logarithm (log)

4.4.8.4.1 Discussion

The log element is the operator that returns a
logarithm to a given base. The base may be specified using a logbase element, which should be the first element
following log, i.e. the second child of the
containing apply element. If the logbase element is not present, a default base of 10 is
assumed.

The log element takes the definitionURL and encoding attributes, which can be used to override the
default semantics.

4.4.9.7.2 Example

4.4.9.7.3 Default Rendering

4.4.10 Linear Algebra

4.4.10.1 Vector (vector)

4.4.10.1.1 Discussion

vector is the container element for a
vector. The child elements form the components of the vector.

For purposes of interaction with matrices and matrix multiplication,
vectors are regarded as equivalent to a matrix consisting of a single
column, and the transpose of a vector behaves the same as a matrix
consisting of a single row. Note that vectors may
be rendered either as a single column or row.

4.4.10.5.2 Example

4.4.10.5.3 Default Rendering

4.4.10.6 Selector (selector)

4.4.10.6.1 Discussion

The selector element is the operator for
indexing into vectors matrices and lists. It accepts one or more
arguments. The first argument identifies the vector, matrix or list from
which the selection is taking place, and the second and subsequent
arguments, if any, indicate the kind of selection taking place.

When selector is used with a single argument, it
should be interpreted as giving the sequence of all elements in the list,
vector or matrix given. The ordering of elements in the sequence for a
matrix is understood to be first by column, then by row. That is, for a
matrix ( ai,j), where the indices
denote row and column, the ordering would be a1,1,
a1,2, ... a2,1, a2,2 ... etc.

When three arguments are given, the last one is ignored for a list or vector,
and in the case of a matrix, the second and third arguments specify the row
and column of the selected element.

When two arguments are given, and the first is a vector or list, the
second argument specifies an element in the list or vector. When a matrix
and only one index
i is specified as in

4.4.10.6.2 Example

4.4.10.6.3 Default Rendering

The selector construct renders in a manner that indicates
which sub-element of the parent object is selected. For vectors and matrices this is
normally done by specifying the parent object together with subscripted indices.
For example, the selection

<apply>
<selector/>
<ci type="vector">V</ci>
<cn> 1 </cn>
</apply>

would have a default rendering of

Selecting the (1,2) element of a 2 by 2 matrix would have a default rendering as

4.4.10.7 Vector product (vectorproduct)

4.4.10.7.1 Discussion

The vectorproduct is the operator element for
deriving the vector product of two vectors.

The vectorproduct element takes the attributes
definitionURL and encoding that can be used to override
the default semantics.

4.4.10.9.2 Example

4.4.10.9.3 Default Rendering

4.4.11 Semantic Mapping Elements

This section explains the use of the semantic mapping elements semantics, annotation and annotation-xml.

4.4.11.1 Annotation (annotation)

4.4.11.1.1 Discussion

The annotation element is the container element
for a semantic annotation in a non-XML format.

The annotation element takes the attributes
definitionURL and encoding that can be used to override
the default semantics. Only the encoding attribute is required whenever
the semantics remains unchanged.

4.4.11.1.2 Example

The annotation element is a semantic mapping
element. It is always used with semantics.

4.4.11.1.3 Default Rendering

None. The information contained in annotations may optionally be used by
a renderer able to process the kind of annotation given.

4.4.11.2 Semantics (semantics)

4.4.11.2.1 Discussion

The semantics element is the container element
that associates additional representations with a given MathML
construct. The semantics element has as its first
child the expression being annotated, and the subsequent children are the
annotations. There is no restriction on the kind of annotation that can be
attached using the semantics element. For example, one might give a TEX
encoding, computer algebra input, or even detailed
mathematical type information in an annotation. A definitionURL attribute is used on
the annotation to indicate when the semantics of an annotation differs significantly from that of the
original expression.

The representations that are XML based are enclosed in an annotation-xml element while those representations that
are to be parsed as PCDATA are enclosed in an annotation element.

The semantics element takes the definitionURL and encoding attributes,
which can be used to reference an external source for some or all of the semantic information.

An important purpose of the semantics construct
is to associate specific semantics with a particular presentation, or
additional presentation information with a content construct. The default
rendering of a semantics element is the default
rendering of its first child. When a MathML-presentation annotation is
provided, a MathML renderer may optionally use this information to render
the MathML construct. This would typically be the case when the first child
is a MathML content construct and the annotation is provided to give a
preferred rendering differing from the default for the content
elements.

4.4.11.2.3 Default Rendering

The default rendering of a semantics element is
the default rendering of its first child.

4.4.11.3 XML-based annotation (annotation-xml)

4.4.11.3.1 Discussion

The annotation-xml container element is used to
contain representations that are XML based. It is always used together with
the semantics element.

The annotation-xml element takes the attributes
definitionURL and encoding that can be used to override
the default semantics. Only the encoding attribute is required whenever
the semantics remains unchanged.